I've seen the following situation modeled using a Binomial RV:
We know that 20% of the people living in a village are against building an airport while 80% are in favor of it. If we randomly pick a sample of 10 people in the village, what is the probability that 3 of them are against building the airport?
I understand the formula $10 \choose 3$ $(0.2)^3 (0.8)^7$
Here's what I'm confused about: Each trial in a binomial experiment has to have the same probability of success/failure.
Let's say we have our 10 randomly selected people in a room (we don't know yet who they are); we ask one to come out first and it turns out it's one of the people against building the airport.
Doesn't that affect the probability that the rest of the randomly selected people in that room are from either group? After all there's one fewer person in the group against to choose from.
Because we're talking about people that aren't being replaced, it seems to me that we're selecting (choosing) people without replacement, so that the probability is never constant from trial to trial.
In that sense, wouldn't it be incorrect to model this situation using a binomial RV?